The hyperboloid at spatial infinity. In an attempt to
overcome the difficulties posed by the fact that the classic
definition of asymptotic simplicity (see Definition
REF  above) makes no reference to the
the behaviour of the gravitational field at spatial infinity, in
[1]} Ashtekar & Hansen introduced a new definition of
asymptotic simplicity in both null and spacelike directions —see
also [2]}. This
definition (asymptotically empty and flat at null infinity and
spatial infinity, AEFANSI) combines the conditions on the conformal extension
given in the definition of asymptotical flatness with further
assumptions on the conformal factor at spatial infinity. Essentially,
it is further required that spatial infinity admits a so-called
point compactification —see, e.g. [3]}, Section 11.6 for a discussion of this notion. The strategy behind the approach
put forward in [1]} is to make use of spacetime notions rather than, say, making use of the initial value
problem. Their definition of asymptotic flatness allows them to blow
up \(i^0\)  to the timelike unit 3-hyperboloid —the hyperboloid
at spatial infinity. The sections of the
hyperboloid give the blowing up of \(i^0\)  to 2-spheres while time direction along the (timelike) generators of the hyperboloid can be,
roughly speaking, associated with all the possible ways in which the
asymptotic region of a Cauchy hypersurface can be boosted. The
definition of AEFANSI spacetimes is geared towards the discussion of
asymptotic symmetries and conserved quantities at spatial infinity.
